Exact eigenvalue order statistics for the reduced density matrix of a bipartite system

Abstract

We consider the reduced density matrix ρa(m) of a bipartite system AB of dimensionality mn (n⩾m without loss of generality) in a Gaussian ensemble of random, complex pure states of the composite system. For a given dimensionality m of the subsystem A, the eigenvalues λ1(m),…,λm(m) of ρa(m) are correlated random variables because their sum equals unity. The following quantities are known, among others: The joint probability density function (PDF) of the eigenvalues λ1(m),…,λm(m) of ρa(m), the PDFs of the smallest eigenvalue λmin(m) and the largest eigenvalue λmax(m), and the family of average values 〈Tr(ρa(m))q〉 parametrized by q. Using these as inputs, we find the exact eigenvalue order statistics for any arbitrary value of m and n, i.e., explicit analytic expressions for the PDFs of each of the m eigenvalues arranged in ascending order from the smallest to the largest one. For the sake of clarity, we first present the eigenvalue order statistics for values of m running from 2 to 6, before going on to the general expressions. When m=n (respectively, m<n) these PDFs are polynomials of order m2−2 (respectively, mn−2) with support in specific sub-intervals of the unit interval, demarcated by appropriate unit step functions. Our exact results are fully corroborated by numerically generated histograms of the ordered set of eigenvalues corresponding to ensembles of over 105 random complex pure states of the bipartite system.